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Fundamental theorem of algebra
The fundamental theorem of algebra, also called D'Alembert's theorem, states that every nonconstant polynomial with complex coefficients has at least one complex root; a degree n polynomial has exactly n.
Fundamental theorem of arithmetic
The fundamental theorem of arithmetic, also called the unique factorization theorem, states that every integer greater than 1 factors into primes in only one way.
Fundamental theorem of Galois theory
The fundamental theorem of Galois theory is a field theory theorem giving an inclusion-reversing correspondence between intermediate fields of a Galois extension and subgroups of its Galois group.
Fused multiply-add
A fused multiply-add (FMA) is a floating-point operation that computes a × b + c with a single rounding, added to IEEE 754 in 2008.
Fusion rules
Fusion rules are non-negative integers that record how two representations of a chiral algebra, such as a vertex operator algebra, combine, given by dimensions of intertwining operator spaces.
Fuzzy arithmetic
Fuzzy arithmetic is the extension of arithmetic operations from real numbers to fuzzy numbers, used to calculate with non-probabilistic uncertainty such as expert judgment or linguistic scales.
Fuzzy number
A fuzzy number is a fuzzy set on the real line representing an imprecise quantity such as "about 5", formulated by Didier Dubois and Henri Prade in 1978.
Galois cohomology
Galois cohomology is the cohomology theory of Galois groups, the right derived functors of the invariants functor, central to class field theory, the Brauer group, and Wiles's proof of Fermat's last theorem.
Galois connection
A Galois connection, also known as a Galois correspondence, is a pair of monotone maps between partially ordered sets where f(p) ≤ q holds exactly when p ≤ g(q), named for Évariste Galois.
Galois cover
A Galois cover is a cover of curves or schemes whose function field extension is a Galois extension, equivalently one with a transitive group of deck transformations.
Galois extension
A Galois extension is an algebraic field extension that is both normal and separable, equivalently the splitting field of a separable polynomial over the base field.
Galois group
A Galois group is the group of automorphisms of a field extension fixing the base field, introduced by Évariste Galois to study when equations are solvable by radicals.
Galois module
A Galois module, also known as a Galois representation, is an abelian group with a compatible action of a Galois group; the term also names the theory of rings of integers.
Galois representation
A Galois representation is a continuous homomorphism from the absolute Galois group of a field to invertible matrices, central to Wiles's proof of Fermat's last theorem.
Galois theory
Galois theory is the branch of algebra, created by Évariste Galois, that associates to each field extension a Galois group and proves that general polynomial equations of degree 5 and higher cannot be solved by radicals.
Gamma matrices
The gamma matrices, or Dirac matrices, are square matrices satisfying the Clifford anticommutation relation {γ, γ} = 2η, appearing as coefficients of the Dirac equation in relativistic quantum mechanics.
GAP (computer algebra system)
GAP (Groups, Algorithms and Programming) is a free, open-source, extensible system for computation in discrete abstract algebra, emphasizing computational group theory and used in research and teaching.
Gauss sum
A Gauss sum, or Gaussian sum, is a finite sum pairing a multiplicative character with an additive character, studied by Carl Friedrich Gauss in 1801.
Gauss–Newton algorithm
The Gauss–Newton algorithm, or Gauss–Newton method, is an iterative method for minimizing nonlinear least squares problems, solving a linear least squares step each iteration without second derivatives.
Gauss–Seidel method
The Gauss–Seidel method is an iterative method for solving linear systems, a refinement of the Jacobi method described by Ludwig Seidel in 1874 and named for him and Carl Friedrich Gauss.
Gaussian approximation (density evolution)
The Gaussian approximation is a one-dimensional simplification of density evolution, assuming decoder messages follow Gaussian distributions; introduced for LDPC codes by Chung, Richardson, and Urbanke in 2001.
Gaussian elimination
Gaussian elimination, also known as row reduction, is an algorithm that solves systems of linear equations by reducing a matrix to upper triangular form and applying back substitution.
Gaussian integer
A Gaussian integer, also known as a complex integer, is a complex number a + bi where a and b are ordinary integers, named for Carl Friedrich Gauss.
Gaussian orthogonal ensemble
The Gaussian orthogonal ensemble (GOE) is a probability distribution on real symmetric matrices whose law is invariant under conjugation by orthogonal matrices, introduced by Eugene Wigner for nuclear level statistics.
Gaussian prime
A Gaussian prime is a Gaussian integer like 1 + i or 3 that cannot be factored except by units, classified by Carl Friedrich Gauss in 1832.
Gaussian symplectic ensemble
The Gaussian symplectic ensemble (GSE) is a random matrix ensemble introduced by Freeman Dyson in 1962 for quantum systems with time-reversal invariance but no rotational symmetry.
GCD domain
A GCD domain, also known as an HCF domain, is an integral domain in which every pair of elements has a greatest common divisor, a generalization of unique factorization domains.
Gelfand representation
The Gelfand representation, also called the Gelfand transform or Gelfand duality, sends each element of a commutative Banach algebra to a function on its character space, introduced by Gelfand in 1941.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem, also called the Gelfand–Naimark representation theorem, is a 1943 result stating that every commutative C-algebra is a function algebra C₀(X) and every C-algebra is a -algebra of bounded operators on a Hilbert space.
Gelfand–Naimark–Segal construction
The Gelfand–Naimark–Segal construction, also called the GNS construction or GNS theorem, converts a state on a C-algebra into a Hilbert-space representation with a cyclic vector, a correspondence Segal completed in 1947.